TY - JOUR
T1 - On the number of union-free families
AU - Balogh, József
AU - Wagner, Adam Zsolt
N1 - Publisher Copyright:
© 2017, Hebrew University of Jerusalem.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2017/4/1
Y1 - 2017/4/1
N2 - A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most (1+o(1))(
n/2
n). Later, Burosch–Demetrovics–Katona–Kleitman–Sapozhenko asked for the number α(n) of such families, and they proved that 2(nn/2)≤α(n)≤222(nn/2)(1+o(1)) They conjectured that the constant 22 can be removed in the exponent of the right-hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.
AB - A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most (1+o(1))(
n/2
n). Later, Burosch–Demetrovics–Katona–Kleitman–Sapozhenko asked for the number α(n) of such families, and they proved that 2(nn/2)≤α(n)≤222(nn/2)(1+o(1)) They conjectured that the constant 22 can be removed in the exponent of the right-hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.
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U2 - 10.1007/s11856-017-1486-y
DO - 10.1007/s11856-017-1486-y
M3 - Article
VL - 219
SP - 431
EP - 448
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
SN - 0021-2172
IS - 1
ER -